354 ON THE DEFINITE INTEGRAL OF THE PRODUCT 



Also fs^dp + P S m P n dp = \S m sJ- 



Jo Jo Jo . 



and S m , S n , being of odd dimensions in p, will vanish when ju, = 0, and 

 they also vanish when /j. = I ; 



Next suppose m and n to be both odd, m being >n+l. By what 

 has been before proved, since m + I and m 1 are even, we may prove that 



p 



J Q 



_ 

 t /* ; 



n- 2r 



2 

 (8), 



for all values of r from 1 to -- . 

 Also 



n n r -11 /_ i 1 ) a 



S n P M ^+ -S M P n ^= , =^^ 

 Jo Jo L Jo 



L \ ,_ 



2 



ri n 



Hence S n P m du is found from S m P n dp. by changing the sign and 

 Jo Jo 



adding another term at the beginning, i.e. taking all values of r from to 



2 ' 



12. Having expressed the values of 



fi 



Jo 



fi 

 P m P n dp and S m P n dft 



o Jo 



in series, we will now determine their values in the form of a single term. 



The theory of these operations may perhaps be presented in a still 

 more simple form. 



First, suppose m to be even and n odd, and let m be the greater. 



Thei if j r (r ' ) - 1 -3.5...(m-r-2) 1 . 3 . 5 ... (n + r- 1) 

 / ^~2.4.6...(">n + r+l) "gTiTGT^n - r) ' 



we 



/"I m+n+l 



have I PP.fc.(-l) "{/(I) +/(3)+/(5) + &c. +/()}. 



It is to be observed that the operation denoted by /, as above defined, 

 has no meaning unless the subject of the operation is an odd integer. 



